A nine-year veteran, Escobar spent just one season in Washington, batting .314 with nine home runs and 56 RBIs.
Though primarily a third baseman, Escobar can play multiple infield positions, giving the Angels some flexibility should they re-sign third baseman David Freese.
In Gott, the Nationals get a 23-year-old right-hander who went 4-2 with a 3.02 ERA in 48 relief appearances last season for Los Angeles.
Washington further bolstered its bullpen on Tuesday by adding free-agent reliever Shawn Kelley on a three-year, $15 million deal, a baseball source told ESPN's Jerry Crasnick.
In 53 appearances for the San Diego Padres last season, Kelley went 2-2 with a 2.45 ERA.